Welcome back to Physics 215. Review gravity Oscillations Simple harmonic motion

Transcription

1 Welcome back to Physics 215 Review gravity Oscillations Simple harmonic motion Physics 215 Spring 2018 Lecture

2 Final Exam: Friday May 4 th 5:15-7:15pm Exam will be 2 hours long Have an exam buddy who can call you. In the physics 208 (here!) Cumulative! Material covered: Textbook chapters 1-11,13,15-17 (weighted towards second half of the semester) Lectures -- all (slides online) Recitation Activities -- all Homework assignments all Midterm Exams 1, 2 & 3 (solutions online) As with Exams 1, 2 & 3, Final is closed book, but you should bring a calculator. A formula sheet will be provided, and it will be available online later this week. Final review session will be in-class Extra office hours for the final will be held next week: times and locations announced next lecture. Physics 215 Spring 2018 Lecture

3 Recall: Energy conservation Consider mass moving in gravitational field of much larger mass M Energy conservation DE = 0, where E = K+U = 1/2mv 2 - GmM/r Hence work done by gravitational field changes kinetic energy: W = -DU = DK Notice E < 0 if object bound Physics 215 Spring 2018 Lecture

4 Escape speed Object can just escape to infinite r if E=0 à (1/2)mv esc 2 = GM E m/r E à v esc 2 = 2GM E /R E Magnitude? 1.1x10 4 m/s on Earth What about on the moon? Sun? Physics 215 Spring 2018 Lecture

5 Consequences for planets Planets with large escape velocities can retain light gas molecules, e.g. Earth has an atmosphere of oxygen, nitrogen Moon does not Conversely Jupiter, Sun manage to retain hydrogen Physics 215 Spring 2018 Lecture

6 Black Holes Suppose light travels at speed c Turn argument about is there a value of M/R for some star which will not allow light photons to escape? Need M/R = c 2 /2G è density = kg/m 3 for object with R = 1m approx Need very high densities possible for collapsed stars Physics 215 Spring 2018 Lecture

7 Review: How to solve gravity problems For circular motion under gravity (i.e. orbits) use: F cent = F g mv 2 /r = GMm/r 2 v here is velocity in the radial direction (tangent to orbit) For escape velocity (or shooting stuff up into space) use: K 1 + U 1 = K 2 + U 2, with U = -GMm/r and K = ½ mv 2 V here is the total velocity (often perpendicular to surface.) Physics 215 Spring 2018 Lecture

8 Gravity Sample problem: The space shuttle, with a mass of 5 million kg, in a 300-km-high orbit wants to capture a smaller satellite (mass of 5,000 kg) for repairs. What is the speed of the shuttle? What is the speed of the satellite? Physics 215 Spring 2018 Lecture

9 Sample problem: A less-than-successful inventor wants to launch small satellites into orbit by launching them straight up from the surface of the earth at a very high speed. A) with what sped should he launch the satellite if it is to have a speed of 500 m/s at a height of 400 km? Ignore air resistance. B) By what percentage would you answer be in error if you used a flat earth approximation (i.e. U = mgh)? Physics 215 Spring 2018 Lecture

10 Oscillations Restoring force leads to oscillations about stable equilibrium point Consider a mass on a spring, or a pendulum Oscillatory phenomena also in many other physical systems... Physics 215 Spring 2018 Lecture

11 Simple Harmonic Oscillator F x = - k x F F F=0 F Spring constant Newton s 2 nd Law for the block: 0 x m a = F x x Differential equation for x(t) d! dx " m $ % = k dt & dt ' x Physics 215 Spring 2018 Lecture

12 Simple Harmonic Oscillator Differential equation for x(t): Solution: d! dx " m $ % = k dt & dt ' x Physics 215 Spring 2018 Lecture

13 Demo: relationship between circular and simple harmonic motion What does w mean? Circle: w is the angular velocity Period T = 2 p/w Projection is a sinusoidal motion! Simple harmonic motion: w is the angular frequency Also sinusoidal motion with SAME T = 2 p/w Physics 215 Spring 2018 Lecture

14 Simple Harmonic Oscillator initial phase amplitude angular frequency Units: A - m T - s f - 1/s = Hz (Hertz) w - rad/s f frequency Number of oscillations per unit time T Period Time taken by one full oscillation Physics 215 Spring 2018 Lecture

15 Simple Harmonic Oscillator DEMO Physics 215 Spring 2018 Lecture

16 SG A mass oscillates on a horizontal spring with period T = 2.0 s. If the amplitude of the oscillation is doubled, the new period will be s s s s s Slide Physics 215 Spring 2018 Lecture

17 SG A block of mass m oscillates on a horizontal spring with period T = 2.0 s. If a second identical block is glued to the top of the first block, the new period will be s s s s s. Slide Physics 215 Spring 2018 Lecture

18 Simple Harmonic Oscillator E = U + K 1 1 E = k x + m v Total Energy dx v = dt E = Physics 215 Spring 2018 Lecture

19 Simple Harmonic Oscillator -- Summary If F = - k x then 1 f = T E = Physics 215 Spring 2018 Lecture k A 2

20 SG A block oscillates on a very long horizontal spring. The graph shows the block s kinetic energy as a function of position. What is the spring constant? 1. 1 N/m N/m N/m N/m. 5. I have no idea. Slide Physics 215 Spring 2018 Lecture

21 Importance of Simple Harmonic Oscillations For all systems near stable equilibrium F net ~ - x where x is a measure of small deviations from the equilibrium All systems exhibit harmonic oscillations near the stable equilibria for small deviations Any oscillation can be represented as superposition (sum) of simple harmonic oscillations (via Fourier transformation) Many non-mechanical systems exhibit harmonic oscillations (e.g., electronics) Physics 215 Spring 2018 Lecture

22 (Gravitational) Pendulum Simple Pendulum Point-like Object What is the force tangent to the curve? 0 q L F net T m q mg Pointlike size of the object small compared to L F net = -mg sinq For small q (in radians) we can use the small angle approximation: F t = - mg s /L Physics 215 Spring 2018 Lecture

23 SG Two pendula are created with the same length string. One pendulum has a bowling ball attached to the end, while the other has a billiard ball attached. The natural frequency of the billiard ball pendulum is: 1. greater 2. smaller 3. the same as the natural frequency of the bowling ball pendulum. Physics 215 Spring 2018 Lecture

24 SG The bowling ball and billiard ball pendula from the previous slide are now adjusted so that the length of the string on the billiard ball pendulum is shorter than that on the bowling ball pendulum. The natural frequency of the billiard ball pendulum is: 1. greater 2. smaller 3. the same as the natural frequency of the bowling ball pendulum. Physics 215 Spring 2018 Lecture

25 (Gravitational) Pendulum Physical Pendulum Extended Object T q d t net = -d mg sinq, d is distance from pivot to center of mass For small q : sinq q m, I t net = -d mg q q mg DEMO Physics 215 Spring 2018 Lecture

26 Sample problem: A physical pendulum consists of a uniform rod of length 0.90 m and mass M = 5 kg, hinged at one end. What is the period of its oscillations? Physics 215 Spring 2018 Lecture

27 Torsion Pendulum (Angular Simple Harmonic Oscillator) t = - k q Torsion constant I d dt # dθ $ & ' ( dt ) = κ θ Solution: Physics 215 Spring 2018 Lecture